The purpose of this research study is to test whether giving T-cells (type of white blood cell that are also known as immune cells) that have been specially processed in the laboratory will help chronic lymphocytic leukemia (CLL) patients' immune system return to normal faster after chemotherapy. This research study will also look into the ability of the lab to process the T-cells for infusion and the side effects of giving T-cells to patients with chronic lymphocytic leukemia (CLL).